Question

If \(S = \{z \in C : |z – i| = |z + i| = |z–1|\}\), then, n(S) is:

• A. 1
• B. 0
• C. 3
• D. 4

Answer 1

The Correct Option is A

To determine the cardinality of set \(S\), we will proceed systematically.

The given set is:

\(S = \{z \in \mathbb{C} : |z - i| = |z + i| = |z - 1|\}\)

In this context, \(z\) represents a complex number, and \(\mathbb{C}\) denotes the collection of all complex numbers.

Initially, we analyze the condition:

\(|z - i| = |z + i|\)

This equation describes a vertical line in the complex plane that is equidistant from the points \(i\) and \(-i\). Due to the symmetry of the imaginary components of \(i\) and \(-i\), this line is the real axis, characterized by an imaginary part of \(z\) equal to 0. Consequently, \(z = x\), where \(x\) is a real number.

Subsequently, we examine the condition:

\(|z - i| = |z - 1|\)

This condition identifies all points in the complex plane equidistant from \(i\) and \(1\). Geometrically, this locus is the perpendicular bisector of the line segment connecting \(i\) and \(1\).

The points \(i\) and \(1\) correspond to the coordinates \((0,1)\) and \((1,0)\) respectively. Their midpoint is calculated as \(\left(\frac{0+1}{2}, \frac{1+0}{2}\right) = \left(\frac{1}{2}, \frac{1}{2}\right)\).

The equation for the perpendicular bisector, which is a vertical line given that the segment joining \(i\) and \(1\) forms a 45-degree angle with the axes, is:

\(x = \frac{1}{2}\)

By integrating these two conditions, the line defined by \(|z - i| = |z + i|\) is the real axis, and the line defined by \(|z - i| = |z - 1|\) is the vertical line \(x = \frac{1}{2}\). The intersection of these two lines is a singular point.

The intersection results in \(z = \frac{1}{2}\).

Therefore, the count of elements in set \(S\), denoted as \(n(S)\), is:

Hence, the correct answer is:

1